Problem 54 - A Complete Resource Book in Mathematics for JEE Main

Question

One mapping is selected at random from all mappings of the set \(S=\\{1,2,3, \ldots n\\}\) into itself. The probability that it is one one is \(\frac{3}{32}\). Then the value of \(n\) is (A) 3 (B) 4 (C) 5 (D) 6

Step-by-Step Solution

Answer

The value of \( n \) is 4.

Step 1: Understand the Problem

We have a set \( S = \{ 1, 2, 3, \ldots, n \} \). We need to find the probability that a randomly selected mapping from this set to itself is one-to-one. We are given that this probability is \( \frac{3}{32} \). We need to find the value of \( n \).

Step 2: Calculate Total Number of Mappings

The total number of mappings from set \( S \) to itself is \( n^n \) because for each of the \( n \) elements in the domain, there are \( n \) choices in the codomain.

Step 3: Calculate Number of One-to-One Mappings

A mapping is one-to-one (injection) if no two elements in the domain map to the same element in the codomain. The number of such mappings is \( n! \) (the permutation of \( n \) elements).

Step 4: Set Up the Probability Equation

The probability of selecting a one-to-one mapping is given by the ratio of the number of one-to-one mappings to the total number of mappings. So, \( \frac{n!}{n^n} = \frac{3}{32} \).

Step 5: Solve the Equation

We need to solve the equation \( \frac{n!}{n^n} = \frac{3}{32} \) for \( n \). Compute this for each value in the options:- **For \( n = 3 \):** \( \frac{3!}{3^3} = \frac{6}{27} = \frac{2}{9} \)- **For \( n = 4 \):** \( \frac{4!}{4^4} = \frac{24}{256} = \frac{3}{32} \)- **For \( n = 5 \):** \( \frac{5!}{5^5} = \frac{120}{3125} \)- **For \( n = 6 \):** \( \frac{6!}{6^6} = \frac{720}{7776} \)Since the equation holds for \( n = 4 \), this is the correct answer.

Key Concepts

One-to-One Mappings

In mathematics, a one-to-one mapping, also known as an injective function, is an essential concept. It represents a situation where each element in the domain is uniquely mapped to an element in the codomain. This means that no two different elements in the domain share the same element in the codomain. To visualize, think of a set of keys (domain), where each key opens a unique lock (codomain).

• Unique Matches: This injective property ensures that each input has a different output.
• No Duplicates: No two elements in the domain map to the same element in the codomain.

To determine if a mapping is one-to-one, check whether each output is associated with only one input. This uniqueness is essential for some mathematical problems, ensuring distinct results without overlap. In our exercise, we calculated the number of one-to-one mappings by using permutations, which we'll discuss in the next section.

Factorials and Permutations

Factorials and permutations play a crucial role in solving problems involving arrangements. The permutation of a set refers to the different ways its elements can be arranged. When finding one-to-one mappings, we use permutations.

• Factorial Concept: Factorial of a number, denoted by \( n! \), is the product of all positive integers up to \( n \). For example, \( 4! = 4 \times 3 \times 2 \times 1 = 24 \).
• Permutations Explained: In our context, permutations (denoted as \( n! \)) represent the one-to-one mappings of a set onto itself. For a set of size \( n \), the number of one-to-one mappings possible is \( n! \).

These concepts help calculate specific arrangements. In our problem, knowing \( n! \) helped us identify that 24 possible one-to-one mappings corresponded to a specific probability.

Domain and Codomain Mappings

Understanding domain and codomain is key to comprehending mappings. In mathematics, the domain is the set of all possible inputs, and the codomain is the set of all potential outputs. For a function \( f: X \rightarrow Y \), \( X \) represents the domain, and \( Y \) is the codomain.

• Domain Basics: It consists of elements that we are mapping from. In our example, it's \( S = \{1, 2, 3, \ldots, n\} \).
• Codomain Essentials: It includes elements that may be mapped to. For a mapping from a set to itself, the domain and codomain are the same.

Mappings are essential to understand functions and their behavior. Each element from the domain maps to an element in the codomain. When mapping a set onto itself, as in our exercise, each input still needs a unique output to maintain the one-to-one relationship. This principle helps compute the number of such mappings correctly.